We recall that the phase space (see [1]}, Ch. 3) is the set of pairs \((p,v)\)  where \(p\in C\)  and \(v\in S_1\)  is a unit vector such such that \(p+\epsilon v\)  is inside \({\mathcal {T}}\)  for small \(\epsilon >0\) . So \(v\)  varies in a semi-circle depending on \(p\in C\) . (During the proofs we shall see how in our context the phase space gives rise to elliptic algebraic surfaces of various types.) In general by a billiard shot we shall mean a pair \((p,v)\)  either in the phase space or such that \(p\in {\mathcal {T}}^o\) , where \(v\)  is a nonzero vector, often assumed of unit length, representing the direction along which \(p\)  is sent. (So in case \(p\in {\mathcal {T}}^o\)  the vector \(v\)  varies in \(S_1\) .) This shot generates a billard trajectory, consisting of the shot together with the further segments which arise by reflection.
